Modern Stochastics: Theory and Applications

Let $({\xi _{1}},{\eta _{1}})$, $({\xi _{2}},{\eta _{2}}),\dots $ be independent identically distributed ${\mathbb{N}^{2}}$-valued random vectors with arbitrarily dependent components. The sequence ${({\Theta _{k}})_{k\in \mathbb{N}}}$ defined by ${\Theta _{k}}={\Pi _{k-1}}\cdot {\eta _{k}}$, where ${\Pi _{0}}=1$ and ${\Pi _{k}}={\xi _{1}}\cdot \dots \cdot {\xi _{k}}$ for $k\in \mathbb{N}$, is called a multiplicative perturbed random walk. Arithmetic properties of the random sets $\{{\Pi _{1}},{\Pi _{2}},\dots ,{\Pi _{k}}\}\subset \mathbb{N}$ and $\{{\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{k}}\}\subset \mathbb{N}$, $k\in \mathbb{N}$, are studied. In particular, distributional limit theorems for their prime counts and for the least common multiple are derived.

A new class of multidimensional locally perturbed random walks called random walks with sticky barriers is introduced and analyzed. The laws of large numbers and functional limit theorems are proved for hitting times of successive barriers.

Probabilistic properties of vantage point trees are studied. A vp-tree built from a sequence of independent identically distributed points in ${[-1,\hspace{0.1667em}1]^{d}}$ with the ${\ell _{\infty }}$-distance function is considered. The length of the leftmost path in the tree, as well as partitions over the space it produces are analyzed. The results include several convergence theorems regarding these characteristics, as the number of nodes in the tree tends to infinity.